Theorem pwpwab 4004

Description: The double power class written as a class abstraction: the class of sets whose union is included in the given class. (Contributed by BJ, 29-Apr-2021.)

Assertion

Ref	Expression
pwpwab	|- ~P ~P A = { x | U. x C_ A }

Distinct variable group:   x, A

Proof of Theorem pwpwab

Step	Hyp	Ref	Expression
1		vex 2766	. . 3 |- x e. _V
2		elpwpw 4003	. . 3 |- ( x e. ~P ~P A <-> ( x e. _V /\ U. x C_ A ) )
3	1, 2	mpbiran 942	. 2 |- ( x e. ~P ~P A <-> U. x C_ A )
4	3	abbi2i 2311	1 |- ~P ~P A = { x | U. x C_ A }

Syntax hints:   = wceq 1364   e. wcel 2167   {cab 2182   _Vcvv 2763   C_ wss 3157   ~Pcpw 3605   U.cuni 3839

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-v 2765  df-in 3163  df-ss 3170  df-pw 3607  df-uni 3840

This theorem is referenced by:  pwpwssunieq  4005